OFFSET
1,1
COMMENTS
Numbers k such that exactly 3 other numbers m < k share the squarefree kernel A007947(k) = rad(k). See A397742 for further information about classifying numbers in this way.
Superset of A030514 (fourth powers of primes).
In the following partition description, p, q, r are primes with p < q < r when they occur together. Let m = rad(k). Let S be the set of the 3 smaller numbers that share the squarefree kernel, m.
We may partition this sequence into 4 blocks as follows:
A. Numbers k of the form m*p^3 (in A397459),
with S = {m, m*p, m*p^2}.
Includes A030514 (fourth powers of primes).
B. Numbers k of the form m*q,
with S = {m, m*p, m*p^2} (in A398442).
C. Numbers k of the form m*p^2 (in A397458),
with S = {m, m*p, m*q}.
D. Numbers k of the form m*r (in A398864),
with S = {m, m*p, m*q}.
Each squarefree number A005117(j) occurs as m = rad(k) for exactly one term k, specifically for k = A284311(4, j-1). So every squarefree number m corresponds to a set S in exactly one of the blocks.
Smallest term k with j >= 2 distinct prime factors is 3*A002110(j).
LINKS
Michael De Vlieger, Table of n, a(n) for n = 1..10000
EXAMPLE
Table of n, a(n) for select n:
n block a(n)
--------------------------------------------------------
1 A 16 2^4
2 C 24 2^3 * 3
3 B 50 2 * 5^2
4 A 81 3^4
5 B 98 2 * 7^2
6 C 120 2^3 * 3 * 5
7 C 135 3^3 * 5
9 A 176 2^4 * 11
16 B 350 2 * 5^2 * 7
17 B 363 3 * 11^2
32 D 735 3 * 5 * 7^2
35 C 840 2^3 * 3 * 5 * 7
392 C 9240 2^3 * 3 * 5 * 7 * 11
5063 C 120120 2^3 * 3 * 5 * 7 * 11 * 13
86248 C 2042040 2^3 * 3 * 5 * 7 * 11 * 13 * 17
MATHEMATICA
rad[x_] := Times @@ FactorInteger[x][[All, 1]]; Select[Range[1200], Function[{k, r}, Count[Range[k], _?(rad[#] == r &)] == 4] @@ {#, rad[#]} &]
CROSSREFS
KEYWORD
nonn,new
AUTHOR
Michael De Vlieger and Peter Munn, Aug 20 2026
STATUS
approved